Domestic Canonical Algebras and Simple Lie Algebras

For each simply-laced Dynkin graph ∆ we realize the simple complex Lie algebra of type ∆ as a quotient algebra of the complex degenerate composition Lie algebra L(A) 1 of a domestic canonical algebra A of type ∆ by some ideal I of L(A) 1 that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of L(A) 1 /I has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M . Introduction Let A be a finite-dimensional algebra over a finite field k with q elements, and consider the free abelian group H(A) with basis the isoclasses of finite A-modules. Then by Ringel [28] H(A) turns out to be an associative ring with identity, called the integral Hall algebra of A, with respect to the multiplication whose structure constants are given by the numbers of filtrations of modules with factors isomorphic to modules that are multiplied (see 2.1). The free abelian subgroup L(A) of H(A) with basis the isoclasses of finite indecomposable A-modules becomes a Lie subalgebra modulo q − 1 whose Lie bracket is given by the commutator of the Hall multiplication. We call this Lie bracket the Hall commutator. It would be interesting to realize all types of simple (complex) Lie algebras using this Hall commutator. Along this line, Ringel [29] realized the positive part of the simple Lie algebra g(∆) for each Dynkin type ∆. Further Peng and Xiao [22] realized all types of simple Lie algebras by the so-called root categories of finite-dimensional representation-finite hereditary algebras. But the Lie bracket was not completely given by the Hall commutator, because the root category R provides only the positive and the negative parts. The Cartan subalgebra h was given by a subgroup of the Grothendieck group of R over the field Q of rational numbers. The Hall commutator was used to define the Lie bracket only inside R, and when the bracket should not be closed in R, namely when we deal with an indecomposable object X in R of a root α and an indecomposable object Y in R of the root −α, the definition of the bracket [X, Y ] was changed in order to have [X, Y ] ∈ h. In [1] we succeeded to realize general linear algebras and special linear algebras (see also Iyama [17]) by the Hall commutator defined on cyclic quiver algebras. In this realization also the Cartan subalgebra was naturally provided together with the 2000 Mathematics Subject Classification. Primary 16G20; Secondary 17B20, 17B60.